= Affine Euler-Lagrange solution from finite derivative fibers
{title2=$f'(u')=c,\quad\#(f')^{-1}(c)<\infty\implies u'\text{ constant for }u\in C^1$}
For an autonomous scalar integral $\int f(u')$, the <first variation> gives the integrated <Euler-Lagrange equation> $f'(u')=c$. If $u$ is $C^1$ and each fiber of $f'$ is finite, the connected image of the interval under the continuous derivative $u'$ lies in a finite set and is therefore a singleton. The fixed endpoints then determine exactly one affine solution. If $f$ is strictly convex, the strict <Jensen's inequality> applied to $u'$ additionally makes this solution the unique <global minimizer> among fixed-endpoint <Lipschitz functions>. Without <convexity> the same unique smooth stationary solution can be a maximizer, for example at zero for $f(v)=1/(1+v^{2008})$ and zero endpoint values.
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